By Theorem REF  if the centralizer is nontrivial, it equals \(\mathcal {C}(L_{2p})={\bf C}[L_{2p},X_p]\)  with \(X_p\)  the operator of minimal order \(p(2s+1)\) , \(s\ne 0\) , in \(\mathcal {C}(L_{2p})\) .
Observe that for \(p=3\)  this is the Fourier transform of Dixmier's example (REF ) [1]}. In this case by Theorem REF  the centralizer is nontrivial and \(X_3\)  has order 9. The pair \(L\) , \(B=X_3\)  is true rank 3.

